We construct non-uniform convergent lattices Γ of pinched, negatively curved Hadamard spaces, in any dimension N≥2. These lattices are exotic, by which we mean that they have a maximal parabolic subgroup P<Γ such that δ(P)=δ(Γ). We also give examples of divergent, non-uniform exotic lattices in dimension N=2. Finally, we consider a particular class of such exotic lattices, with infinite Bowen–Margulis measure and whose cusps have a particular asymptotic profile (satisfying a “heavy tail condition”), and we give precise estimates of their orbital function; namely, we show that their orbital function is lower exponential with asymptotic behaviour ≍eδΓRR1-κL(R), for a slowly varying function L.